David Lidington said he believed there would be a Commons majority for a Brexit deal.

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David Lidington (Peter Byrne/PA)

Agreement on a Brexit deal with Brussels will create a “new dynamic” in Parliament which would help secure support for Theresa May’s plans, a key ally of the Prime Minister has claimed.

The fragile alliance keeping the Prime Minister in power has been strained as the Democratic Unionist Party railed against measures it fears will create a border down the Irish Sea between Northern Ireland and the rest of the UK.

But David Lidington, Mrs May’s de facto deputy prime minister, said he hoped that once a deal was on the table MPs would rally behind it.

The PM’s letter raises alarm bells for those who value the integrity of our precious union & for those who want a proper Brexit for the whole UK. From her letter, it appears the PM is wedded to the idea of a border down the Irish Sea with NI in the EU SM regulatory regime. https://t.co/qN6c41zx7U

Hopes of an imminent breakthrough in the Brexit negotiations have so far failed to be met.

But both Mr Lidington and Ireland’s Leo Varadkar suggested a deal could be reached between the UK and EU in the coming weeks.

The Taoiseach said: “A successful outcome is not guaranteed but I think it is possible in the next couple of weeks.”

Leo Varadkar said a deal could be done in the next few weeks but ‘lots of things can go wrong’ (Peter Byrne/PA)

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Tensions between Mrs May and her DUP allies have been exposed amid concerns about measures aimed at avoiding a hard border on the island of Ireland.

DUP leader Arlene Foster said the Prime Minister appeared “wedded to the idea of a border down the Irish Sea” despite Downing Street’s repeated assurances to the contrary.

The response of the DUP has caused frustration in Downing Street, with sources insisting that Mrs May was not hiding behind “weasel words” and had stressed that she would not accept a deal which saw Northern Ireland hived off.

A leaked letter from the Prime Minister in reply to an earlier message from Mrs Foster and her deputy, Nigel Dodds, set out Mrs May’s approach.

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She wants a “backstop” measure which would create a temporary “joint customs territory” with the EU for the whole of the UK.

But Brussels appears set to insist on a Northern Ireland-only “backstop to the backstop” in case negotiations on a wider UK approach break down or any time limit on it expires.

(PA Graphics)

In the letter, obtained by the Times, Mrs May said: “I am clear that I could not accept there being any circumstances or conditions in which that ‘backstop to the backstop’, which would break up the UK customs territory, could come in to force.”

But she acknowledged that the “unique circumstances” of Northern Ireland “could require specific alignment solutions in some scenarios” on regulations.

The scope of any alignment with Brussels’ rules would be limited to what is “strictly necessary” to avoid a hard border.

The DUP has interpreted the wording of her letter to mean that specific Northern Ireland measures will be contained in the Brexit divorce deal despite Mrs May’s assurances.

Theresa May depends on the 10 MPs from Arlene Foster’s DUP to support her minority administration (Clodagh Kilcoyne/PA)

Mrs Foster said: “It appears the Prime Minister is wedded to the idea of a border down the Irish Sea with Northern Ireland in the EU single market regulatory regime.”

“Our position has always been and remains that we will not support arrangements that leave Northern Ireland separated from the rest of the United Kingdom and tied to the European Union’s customs or regulatory regimes.” https://t.co/J8ywSuhK8c

DUP Brexit spokesman Sammy Wilson told BBC Radio 4’s Today programme “we want to trust the Prime Minister” but “you have to judge any promise by what is actually delivered in an agreement”.

The party’s Westminster leader, Mr Dodds, warned on Sky News: “I think the Prime Minister will hopefully realise what can be got through parliament and what can’t.”

Any version of the backstop would apply unless and until a wider UK-EU deal on the future relationship solved the issue of how to avoid a hard border with Ireland.

The Prime Minister relies on the support of the DUP’s 10 MPs for her Commons majority, votes which will be crucial as she attempts to get a deal through Parliament.

At the British-Irish Council summit in the Isle of Man, Mr Lidington said: “The Prime Minister has always been very clear we won’t accept something that involves carving out Northern Ireland from the rest of the UK.”

He said a UK-EU deal would involve “compromises, give and take on all sides” but when faced with “product on the table” in the form of an agreement backed by all 28 governments there could be a shift in attitude at Westminster.

“People will need to ask themselves what is it that is going to be in the best interests of those who sent them to Westminster to represent them, to ensure that we maintain living standards and investment and prosperity and employment in our country.

“I hope and I believe that we can secure that majority in Parliament for the agreement.”

Mr Varadkar, who was also at the Isle of Man summit, said “it is more likely than not that we will be able to conclude an agreement” in the coming weeks but “lots of things can go wrong” – including having to get a deal through Westminster and the European Parliament.

“Even when all of that is done, then we begin the talks on the future relationship,” he added.

“There is no clean break here, Brexit is going to go on for a very long time.”